Elementary recursive bounds for Hilbert’s
17-th problem: why 5 levels of exponentials ?
by Marie-Françoise Roy
IRMAR
CNRS/Université de Rennes 1
This talk is based on joint work with Henri Lombardi (Besançon, France) and
Daniel Perrucci (Buenos Aires, Argentina). Follows a first talk by Daniel.
1
Section 1
2
1 Sums of squares
Hilbert’s question (1900), X = (X1, Xk): is it true that
P ∈ R[X] is non negative if and only if P ∈
P
R(X) 2
sum of squares of rational function (with denominators)
Artin’s positive answer (1927): abstract algebra
Result about the reals, method of proof uses much more abstract objects :
real closures of the field of rational functions.
Sums of squares
3
Outline of Artin’s proof:
•
Start from P which is not a sum of squares of rational functions.
•
Sums of squares do not contain P and form a proper cone.
•
Using Zorn’s lemma, obtain a maximal proper cone which does not contain
P . Such a maximal cone is a total order.
•
Taking the real closure of the field of rational functions for this order, get
a field in which P takes negative values (the images of the variables).
•
If P takes negative values in a real closed field containing the reals, P
takes negative values over the reals. This is the first instance of a transfer
principle in real algebraic geometry. Based on Sturm’s theorem.
“quite indirect” : by contradiction + use of Zorn
Section 1
4
Many problems after Artin’s proof.
Effectivity problems : is there an algorithm checking whether a given polynomial is everywhere nonnegative and if so provides a representation as a sum of
squares?
Complexity problems : what are the best possible bounds on the degrees of
the polynomials in this representation?
Former results (see Daniel’s talk for references to former work): primitive
recursive degree bounds.
Our work [LPR] gives elementary recursive degree, precisely (d degree, k
number of variables of the input polynomial)
22
4
2d
k
Sums of squares
5
Strategy For every unrealizable H, construct an algebraic identity ↓H↓ and
control the degrees for the Positivstellensatz. Hilbert 17 th problem follows (cf
talk by Daniel).
Principle of the proof Translate a (very elementary) proof that H is unrealizable into an incompatibility (algebraic identity) ↓H↓.
Method already used by Lombardi in 1990 [Lom] to give primitive recursive
bounds : the construction of the impossibility follows the proof of unrealizability based on the very elementary proof produced by the quantifier elimination
method of Hormander [Hor],[BCR].
primitive recursive degree bounds coming from Hormander’s
quantifier elimination algorithm.
6
Section 1
Here the construction of the incompatibility uses more sophisticated tools that
can be used to prove the unrealizability
1. a real polynomial of odd degree has a real root
2. a real polynomial has a complex root (algebraic proof due to Laplace)
3. number of real roots given by the signature of Hermite’s quadratic form
(moment matrix (!)) which is also determined by sign conditions on principal minors
4. Sylvester’s inertia law: the signature of a quadratic form is well defined
5. cylindrical decomposition : realizable sign conditions for P ⊂ R[X1, , Xk]
fixed by list of non empty sign conditions for ProjXk(P) ⊂ R[X1, , Xk−1].
ProjXk(P) contains minors of several Hermite quadratic forms (using
Thom’s encoding of real roots by sign of derivatives)
necessary to construct the corresponding incompatibilities, controling the
degrees.
main concept : weak inference (see Daniel’s talk)
basic tools: algebraic identities, case by case reasoning (see Daniel’ talk)
Sums of squares
7
1.1 Base case: univariate polynomial (with parameters
A polynomial of odd degree has a real root
P (u, y) polynomial monic with respect to y, of odd degree in y (with parameters u (many))
Since P has a root in R for every u ∈ Rk it is clear that, for every system of
sign conditions H (“context”)
if H ∧ P (u, t) = 0 is unrealizable, then H is unrealizable
We need to prove
⊢ ∃t P (u, t) = 0
i.e. give a construction of ↓H↓ starting from ↓H, P (u, t) = 0↓
“goes from right to left”
Section 1
8
Artin-Shreier’s [AS] : in a maximally real field (-1 is not a sum of squares but
in every algebraic extension -1 a sum of squares), every polynomial of odd degree
has a root: proof is by induction on the degree p, using euclidean division.
Theorem 1. Let P (u, y) ∈ K[u, y] monic with respect to y of odd degree p in y.
Then
⊢ ∃t P (u, t) = 0
Suppose we have an initial incompatibility ↓H, P (u, t) = 0↓ in variables (v, t),
where v ⊃ u and t v, with degree in v bounded by δv and degree in t bounded
by δt. The final incompatibility ↓H↓ has degree in v bounded by 3g1(p) (δv +
δt degu (P )), where
g1(p) = 23·2
p−1
p p.
Degree: adds two level of exponentials
Sums of squares
9
A polynomial with real coefficients has a complex root [Lap]
P (u, t) a polynomial monic and of degree p with respect to T .
P (u, a + i b) = 0 is an abbreviation for the two equalities
PIm(u, a, b) = 0,PRe(u, a, b) = 0
expressing that the real and imaginary part of
P (u, a + i b)
are zero.
Since P has a root in R[i] for every υ ∈ Rk, it is clear that if H ∧ P (u, a + i b) = 0
is unrealizable then H is unrealizable.
Need to prove
⊢ ∃z [ P (z) = 0 ]
i.e. to give a construction of ↓H↓ from ↓H, P (u, a + i b) = 0↓
“goes from right to left”
Section 1
10
Based on the algebraic proof of the fundamental theorem of algebra (Laplace).
Proof by induction on the exponent of 2 in the degree p, constructs a polynomial
of odd degree 6p p, which has a real root, then a complex root of P by successive
quadratic extensions.
Theorem 2. Let P (y) ∈ K[u, y] monic with respect to y, of degree p in y,
p = 2r o ≥ 1, with o odd. Then (z = a + i b)
⊢ ∃z [ P (z) = 0 ]
Suppose we have an initial incompatibility ↓H, P (u, a + i b) = 0↓ in K[v,
a, b], where v ⊃ v and a, b v, degree in v bounded by δv and degree in (a,
b) bounded by δz. The final incompatibility ↓H↓ has degree in v bounded by
g 2(p) (δv + δz degu (P )) where
g2(p) = 2
p r
3( )2
2 2
Degree: adds three level of exponentials
Sums of squares
11
Factorization into irreducible real factors of degree 1 and 2 as incompabilities, with degree bounds.
Fact µ,ν (t, z)
describes the factorization of P into #µ distinct real factors of degree 1 with
multiplities µ1, , µ#µ and #ν distinct irreducible real factors of degree 2
(tk + ak) 2 + bk2
with multiplities ν1, , ν#ν , t = (t1, , t#µ) and z = (z1, , z#ν ), zk = ak + i bk , bk=0.
That two quadratic factors are distinct is expressed by
2
2
′ 2
Res y (y − a) + b , (y − a ) + b
′2
= ((a − a ′)2 + (b − b ′)2) · ((a − a ′)2 + (b + b ′)2)=0
where Res y is the resutant polynomial in the variable y.
Degree: adds three level of exponentials
Section 1
12
Hermite’s method [Her], [BPR]
P (u, t) a polynomial monic in t of degree p.
Hermite’s quadratic form, using Newton sums
X
µ(t)(f1 + f2 t + + f p t p−1)2,
Her(P )(u) =
t∈Z(P (u),R[i])
Hermite’s theory
Rk(Her(P )(u)) = number of complex roots of P (u, t)
Si(Her(P )(u)) = number of real roots of P (u, t)
Complex conjugate roots give a difference of two squares, only real roots contribute to the signature.
Sums of squares
13
fixing the number of distinct real and complex roots with multiplicities of
P by (µ, ν), get a value (RkFact(µ, ν), SiFact(µ, ν)) of the rank and signature of
Hermite’s quadratic form.
fixing a sign condition τ ∈ {−1, 0, 1}0, ,p−1 on the principal minors of Hermite’s quadratic form, get a value (RkHMi(τ ), SiHMi(τ )) of its rank and signature.
By Sylvester’s inertia law,hese two values of the rank and signature cannot
be different.
Theorem 3. P (u, y) ∈ K[u, y]monic with respect to y of degree p in y.
If (RkHMi(τ ), SiHMi(τ )) (RkFact(µ, ν), SiFact(µ, ν)), we have
↓sign(HMiP )) = τ , Fact(P ) µ,ν (t, z)↓
with degree bounded by g H (p) degt,z (P ), with
g H (p) = 21 · 27p+1 p5p+6 34p+2.
Degree: one level of exponentials
Section 1
14
Similar result (statement would be more technical) for a useful generalization
(sign determination) [BPR]
Hermite’s quadratic form Her(P , Q)
X
Her(P , Q) =
µ(t)Q(t)(f1 + f2 t + + f p t p−1)2
t∈Zer(P ,R[i])
Rank(Her(P , Q)) = #{x ∈ Zer(P , R[i])N Q(x)=0}
Sign(Her(P , Q)) = #{x ∈ Zer(P , R)N Q(x) > 0} − #{x ∈ Zer(P , R)N Q(x) < 0}
also determined by signs of principal minors
Sums of squares
15
1.2 Removing one variable: cylindrical decomposition
P s polynomials in k variables, cylindrical decomposition produces ProjXk(P)
in k − 1 variables (minors of Hermite quadratic forms associated to products of
(few) polynomials of P). To every realizable sign condition τ on ProjXk(P) is
associated the list Sign(τ , P) of realizable sign conditions on P implied by τ .
Uses subresultants, sign determination, Thom encodings ...
If H is such that for every σ ∈ Sign(τ , P), H ∧ sign(P) = σ is unrealizable,
then H ∧ sign(ProjXk(P)) = τ is unrealizable. Need to construct, given incompatibilities ↓H ∧ sign(P) = σ↓ for every σ ∈ Sign(τ , P), an incompatibility
↓H ∧ sign(ProjXk(P)) = σ↓ which is the meaning of
_
(⋆) sign(ProjXk(P)) = τ ⊢
sign(P) = σ
σ∈Sign(τ ,P)
“goes from right to left”
Degree: adds three level of exponentials
induction on the number of variables: the degree produced at the end (unik
variate polynomials) ProjX2( ProjXk(P)) are d 4
Section 1
16
1.3 Back to effective Hilbert 17 th problem
If it is true that P > 0 everywhere, want to construct an incompatibility ↓H↓
where H = {P }, H ≥ = {−P }, H= = ∅ (expressing that P < 0 is empty)
start from incompatibility between P > 0 and H
P 2 + (−P ) P = 0 (1)
and incompatibility between P = 0 and H
P 2 + (−P ) P = 0 (2)
using the inference (⋆) for ProjXk({P }) , we get an incompatibility between H
and every relizable sign condition on ProjXk({P })
using the inference (⋆) for ProjXk({P }), we get an incompatibility between H and
every relizable sign condition on ProjXk −1(ProjXk({P }))
....
using k times the inference (⋆) we get an incompability between H and ...
nothing .. because all the variables have been eliminated
Sums of squares
17
as seen before (talk of Daniel) an incompatibility of ↓H↓ where H = {P },
H ≥ = {−P }, H= = ∅ (expressing that P < 0 is empty) is of the form
X
X
2
2e
R j2 P = 0
Qi −
P +
R j2 P 2
P = 2e P 2 =
P +
Qi
P
P
P 2
Qi )
R j2 P 2 (P 2k +
P 22
,
(P 2e +
Qi )
k
degree of the last eliminating family doubly exponential d4 , to
plug in a tower with already three level of exponents: we get
five !
Section 2
18
2 Discussion
Classical Nullstellensatz: single exponential bounds [Bro].
Existential theory of the reals: single exponential bounds (when based on critical
point method, not on CAD, use infinitesimal deformation) [BPR].
Why not single exponential certificate for Hilbert 17th problem ?
Trade-off: “elementary proof’ correspond to’ “long certificate”, “more sophisticated proof” correspond to “shorter certificate”.
Discussion
19
Projects for the near future: finalize the paper (75 pages), first talk announcing
the result in NY in jan 1991 (23 years ago)
Projects for the future
- use another algebraic proof of the existence of complex roots (two
exponentials rather than three) (recent work of M. Eisermann)
- use critical point method rather than projection variable after variable (one
exponential rather than two)
Section
20
Bibliography
[A]
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(1927). The collected papers of Emil Artin, 273-288. Reading: Addison-Wesley (1965).
[AS] E. Artin, O. Schreier, Algebraische Konstruktion reeller Körper , Hamb. Abh. 5 8(99 (1925). The collected papers of Emil Artin, 258-271. Addison-Wesley (1965).
[BPR] S. Basu, R. Pollack, M.-F. Roy, Algorithms in real algebraic geometry,
Springer-Verlag, second edition (2006). Revised version of the first edition on line at
http://perso.univ-rennes1.fr/marie-francoise.roy/
[BCR] J. Bochnak, M. Coste, M.-F. Roy, Real algebraic geometry, Springer-Verlag,
second edition in english (1998)
[Col] G. E. Collins,. Quantifier elimination for real closed fields by cylindric algebraic
decomposition. In Second GI Conference on Automata Theory and Formal Languages,
volume 33 of Lecture Notes in Computer Science, pages 134–183, Berlin, 1975. SpringerVerlag.
Bibliography
[E]
21
M. Eisermann, The Fundamental Theorem of Algebra made effective: an elementary
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[Kr]
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[Kri] J.-L. Krivine, Anneaux préordonnés, J. Analyse Math., 12, 307-326 (1964)
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